By Jan Kok & Warren D.Smith.
(There is also an easier-to-digest[?] RRV page by Ivan Ryan.)

Reweighted Range Voting (RRV)
is based on STV (it uses the STV ideas of "Droop quota" and "ballot
reweighting"), and chooses multiple winners in such a way as to obtain
proportional representation. The main differences between RRV and STV
that would be apparent to ordinary people are:

RRV uses a more-expressive Range-Voting-style
ballot – voters "score" each candidate from (say) 0=bad to 9=good –
while STV uses rank-order ballots in which expressing "strength of preference"
is impossible;

It is simpler to explain how RRV works.

How RRV works

Each ballot is given an initial "weight" of 1.

The weighted scores on the ballots are summed for each candidate,
thus obtaining that candidate's total score.

The candidate with the highest total score (who has not already
won) is declared a winner.
(Note that the first winner in RRV is the
same as the winner of a ordinary single-winner Range Voting
election using the same ballots.)

When a voter "gets her way" in the sense that a candidate she rated
highly wins, her ballot weight should be reduced so that she has less
influence on later choices of winners. To accomplish that, each ballot is given a
new weight = 1/(1+SUM/MAX), where SUM is the sum of the scores
that ballot gives
to the winners-so-far, while MAX is the maximum allowed score
(e.g. MAX=99 if allowed scores are in the range 0 to 99).

Repeat steps b-d until the desired number of winners has
been chosen.

One can instead employ this formula in step d:
weight = K / (K + SUM/MAX)
where K is any positive constant. The range
½≤K≤1 seems most interesting.
Our formula above had used K=1, which is analogous to Jefferson & d'Hondt notions
of proportionality, which tend to favor large political parties (incentivizes smaller
parties to merge); meanwhile
K=½
is analogous to Webster & Sainte-Laguë notions, which tend to be "fairer" and provide
much smaller merge-incentive (and perhaps not even of the same sign, it might be a slight
split-incentive).

We also remark that votes incorporating "no opinion" scores on candidates could also be allowed.
Ballots employing them will not affect the weighted-average scores for any candidates
they rate with "no opinion" and such ballots will not be re-weighted when such
a candidate wins.

Beyond RRV's
obvious simplicity advantage, it has other advantages such as
monotonicity. That is, with RRV, if a voter
increases a rating for a candidate,
that will never change that candidate from a winner to a loser.
(With
STV, giving a candidate a better rank can cause that candidate to
lose,
even in the single-winner [IRV] case.)

You may be thinking, "STV is good enough, why should we consider
another PR method?" One good reason to think about RRV
has to do with single-winner methods.

RRV is a PR method that
doesn't require IRV as a stepping stone. Rather, it uses
Range Voting as the stepping stone; RV is the corresponding
single-winner method.

Proportionality Theorem

If some voter faction (call them the "Reds"),
consisting of a fraction F (where 0≤F<1) of the voters,
wants to, it is capable (regardless of what the other voters do) of electing at
least
⌊(1+N)F-⌋
red winners (assuming, of course, that at least this many red candidates run, and the total
number of winners is to be N).

Specifically, it can accomplish that by voting MAX for all Reds and MIN for everybody else.

To say that again: if 37% of the voters are reds, they can assure at least about 37% red winners
(up to rounding-to-integers effects).

Imperfections

While RRV seems superior to STV both in simplicity and properties,
that is not to say that it is perfect. Two flaws in RRV
(which also are flaws in STV) are

a multiwinner analogue of "participation failure,"
and

the fact that it cannot be "counted in precincts" but
only centrally.
Forest Simmons in 2007 solved an open problem by showing
how to design PR multiwinner voting methods that are
countable in precincts – see puzzle 15 –
but that is another story for another day.

To explain the former:
here is a desirable-sounding property for
multiwinner voting systems:

Multiwinner "participation property":
By casting an honest vote, you cannot cause X to be elected
instead of Q (with all other winners staying the same)
if you prefer Q over X.

The "STV" system used in Ireland and Australia definitely fails this property,
since its single-winner special case (instant runoff voting, IRV)
fails it.

What about our new RRV system? It obeys this property in the single-winner
case (because that is just range voting).
But it fails it in the following
140-voter 3-candidate 2-winner election example:

#voters

their vote

50

Z=99, X=42, Q=0

50

X=99, Q=43, Z=0

40

Q=99, Z=53, X=0

In the first round,
the totals are
Z=7070,
X=7050,
and
Q=6110,
so
RRV elects Z.
That deweights X so that,
second round, Q wins.
(The second round totals are
Q=6110,
and
X=6000.)

It turns out however, that I can argue that every
proportional representation voting method
must fail a property of this ilk; we shall hopefully explain that in a
to-be-written future web page.

The OSCARs

RRV is now used by the OSCARs
to select the 5 nominees for "Best Visual Effects" award
for movies each year, according to this rule

"Five productions shall be selected using reweighted range voting to
become the nominations for final voting for the Visual Effects Award."

from
http://www.oscars.org/awards/academyawards/rules/rule22.html.
They use an 0-10 scale for ratings.
Voting for the OSCARs is run by the accounting firm
PriceWaterhouseCoopers. [Presumably, this was done as an experiment with the idea of switching other
OSCAR categories to use RRV if it worked out well.]
The apparently first time (2013) they employed this system, the 5 nominees were: